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Unformatted text preview: Math 2331 Linear Algebra Section 4.4 Orthonormal Bases and Gram-Schmidt Vectors are orthogonal if their dot product is 0. A set of vectors is called orthogonal if the dot product of any two vectors in the set is 0 Definition: The vectors are orthonormal if 1 2 , ,..., n q q q 1 T i j i j i j q q q q i j ≠ ⎧ • = = ⎨ = ⎩ An orthonormal basis for a vector space is a set of vectors that is both a basis for the space and an orthonormal set. A matrix with orthonormal columns is denoted Q in the textbook. These matrices are VERY NICE. (note that having orthonormal columns does not mean the matrix is square. Important properties of matrices with orthonormal columns 1. T Q Q I = 2. If Q is square, 1 T Q Q − = 3. Qx has the same length a x for all x. Qx x = 4. Q preserves dot products - ( ) ( ) T T Qx Qy x y = Examples of Matrices with orthonormal columns- Rotation Matrices in the plane cos sin sin cos Q θ θ θ θ − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ and 1 cos sin sin cos T Q Q θ θ θ θ...
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